The posts of a hockey goal are 6ft apart. A player attempts to score by shooting the puck from a point that is 19ft from one post and 24ft from the other post. Within what angle (cos) must the shot be made?

Question

anonymous · Answer

If you remember the law of cosines, $$c^2=a^2+b^2-2ab*\cos( \theta)$$
then all you have to do is solve for cos(theta)
$$\cos( \theta)=(c^2-a^2-b^2)/(-2ab)$$
and then plug in the values. This step can be tricky as you have to choose which angle you want. By definition, the cos( theta) that you will get from the law of cosines is the angle formed directly across the triangle from side c. We will thus call the line formed between the goalposts side c.
Solving for cos( theta),
$$\cos( \theta)=(6^2-19^2-24^2)/(-2*29*24)=901/1392$$
If you need to, you can then solve for theta
$$\theta=\arccos(901/1392)\approx.867$$